Acceleration. Uniformly accelerated motion

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Acceleration is a quantity that characterizes the rate of change in speed.

For example, when a car starts moving, it increases its speed, that is, it moves faster. At first its speed is zero. Once moving, the car gradually accelerates to a certain speed. If a red traffic light comes on on its way, the car will stop. But it will not stop immediately, but over time. That is, its speed will decrease down to zero - the car will move slowly until it stops completely. However, in physics there is no term “slowdown”. If a body moves, slowing down its speed, then this will also be an acceleration of the body, only with a minus sign (as you remember, speed is a vector quantity).

> is the ratio of the change in speed to the period of time during which this change occurred. The average acceleration can be determined by the formula:

Rice. 1.8. Average acceleration. In SI acceleration unit– is 1 meter per second per second (or meter per second squared), that is

A meter per second squared is equal to the acceleration of a rectilinearly moving point, at which the speed of this point increases by 1 m/s in one second. In other words, acceleration determines how much the speed of a body changes in one second. For example, if the acceleration is 5 m/s2, then this means that the speed of the body increases by 5 m/s every second.

Instantaneous acceleration of a body (material point) at a given moment in time is a physical quantity equal to the limit to which the average acceleration tends as the time interval tends to zero. In other words, this is the acceleration that the body develops in a very short period of time:

With accelerated linear motion, the speed of the body increases in absolute value, that is

V 2 > v 1

and the direction of the acceleration vector coincides with the velocity vector

If the speed of a body decreases in absolute value, that is

V 2< v 1

then the direction of the acceleration vector is opposite to the direction of the velocity vector. In other words, in this case what happens is slowing down, in this case the acceleration will be negative (and< 0). На рис. 1.9 показано направление векторов ускорения при прямолинейном движении тела для случая ускорения и замедления.

Rice. 1.9. Instant acceleration.

When moving along a curved path, not only the speed module changes, but also its direction. In this case, the acceleration vector is represented as two components (see next section).

Tangential (tangential) acceleration– this is the component of the acceleration vector directed along the tangent to the trajectory at a given point of the movement trajectory. Tangential acceleration characterizes the change in speed modulo during curvilinear motion.

Rice. 1.10. Tangential acceleration.

The direction of the tangential acceleration vector (see Fig. 1.10) coincides with the direction of linear velocity or is opposite to it. That is, the tangential acceleration vector lies on the same axis with the tangent circle, which is the trajectory of the body.

Normal acceleration

Normal acceleration is the component of the acceleration vector directed along the normal to the trajectory of motion at a given point on the trajectory of the body. That is, the normal acceleration vector is perpendicular to the linear speed of movement (see Fig. 1.10). Normal acceleration characterizes the change in speed in direction and is denoted by the letter. The normal acceleration vector is directed along the radius of curvature of the trajectory.

Full acceleration

Full acceleration during curvilinear motion, it consists of tangential and normal accelerations along and is determined by the formula:

(according to the Pythagorean theorem for a rectangular rectangle).

Uniformly accelerated motion is motion with acceleration, the vector of which does not change in magnitude and direction. Examples of such movement: a bicycle rolling down a hill; a stone thrown at an angle to the horizontal.

Let's consider the last case in more detail. At any point of the trajectory, the stone is affected by the acceleration of gravity g →, which does not change in magnitude and is always directed in one direction.

The motion of a body thrown at an angle to the horizontal can be represented as the sum of motions relative to the vertical and horizontal axes.

Along the X axis the movement is uniform and rectilinear, and along the Y axis it is uniformly accelerated and rectilinear. We will consider the projections of the velocity and acceleration vectors on the axis.

Formula for speed during uniformly accelerated motion:

Here v 0 is the initial velocity of the body, a = c o n s t is the acceleration.

Let us show on the graph that with uniformly accelerated motion the dependence v (t) has the form of a straight line.

Acceleration can be determined by the slope of the velocity graph. In the figure above, the acceleration modulus is equal to the ratio of the sides of triangle ABC.

a = v - v 0 t = B C A C

The larger the angle β, the greater the slope (steepness) of the graph relative to the time axis. Accordingly, the greater the acceleration of the body.

For the first graph: v 0 = - 2 m s; a = 0.5 m s 2.

For the second graph: v 0 = 3 m s; a = - 1 3 m s 2 .

Using this graph, you can also calculate the displacement of the body during time t. How to do it?

Let us highlight a small period of time ∆ t on the graph. We will assume that it is so small that the movement during the time ∆t can be considered a uniform movement with a speed equal to the speed of the body in the middle of the interval ∆t. Then, the displacement ∆ s during the time ∆ t will be equal to ∆ s = v ∆ t.

Let us divide the entire time t into infinitesimal intervals ∆ t. The displacement s during time t is equal to the area of ​​the trapezoid O D E F .

s = O D + E F 2 O F = v 0 + v 2 t = 2 v 0 + (v - v 0) 2 t .

We know that v - v 0 = a t, so the final formula for moving the body will take the form:

s = v 0 t + a t 2 2

In order to find the coordinate of the body at a given time, you need to add displacement to the initial coordinate of the body. The change in coordinates during uniformly accelerated motion expresses the law of uniformly accelerated motion.

Law of uniformly accelerated motion

y = y 0 + v 0 t + a t 2 2 .

Another common problem that arises when analyzing uniformly accelerated motion is finding the displacement for given values ​​of the initial and final velocities and acceleration.

Eliminating t from the equations written above and solving them, we obtain:

s = v 2 - v 0 2 2 a.

Using the known initial velocity, acceleration and displacement, the final velocity of the body can be found:

v = v 0 2 + 2 a s .

For v 0 = 0 s = v 2 2 a and v = 2 a s

Important!

The quantities v, v 0, a, y 0, s included in the expressions are algebraic quantities. Depending on the nature of the movement and the direction of the coordinate axes under the conditions of a specific task, they can take on both positive and negative values.

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In this topic we will look at a very special type of irregular motion. Based on the opposition to uniform movement, uneven movement is movement at unequal speed along any trajectory. What is the peculiarity of uniformly accelerated motion? This is an uneven movement, but which "equally accelerated". We associate acceleration with increasing speed. Let's remember the word "equal", we get an equal increase in speed. How do we understand “equal increase in speed”, how can we evaluate whether the speed is increasing equally or not? To do this, we need to record time and estimate the speed over the same time interval. For example, a car starts to move, in the first two seconds it develops a speed of up to 10 m/s, in the next two seconds it reaches 20 m/s, and after another two seconds it already moves at a speed of 30 m/s. Every two seconds the speed increases and each time by 10 m/s. This is uniformly accelerated motion.

The physical quantity that characterizes how much the speed increases each time is called acceleration.

Can the movement of a cyclist be considered uniformly accelerated if, after stopping, in the first minute his speed is 7 km/h, in the second - 9 km/h, in the third - 12 km/h? It is forbidden! The cyclist accelerates, but not equally, first he accelerated by 7 km/h (7-0), then by 2 km/h (9-7), then by 3 km/h (12-9).

Typically, movement with increasing speed is called accelerated movement. Movement with decreasing speed is slow motion. But physicists call any movement with changing speed accelerated movement. Whether the car starts moving (the speed increases!) or brakes (the speed decreases!), in any case it moves with acceleration.

Uniformly accelerated motion- this is the movement of a body in which its speed for any equal intervals of time changes(can increase or decrease) the same

Body acceleration

Acceleration characterizes the rate of change in speed. This is the number by which the speed changes every second. If the acceleration of a body is large in magnitude, this means that the body quickly gains speed (when it accelerates) or quickly loses it (when braking). Acceleration is a physical vector quantity, numerically equal to the ratio of the change in speed to the period of time during which this change occurred.

Let's determine the acceleration in the next problem. At the initial moment of time, the speed of the ship was 3 m/s, at the end of the first second the speed of the ship became 5 m/s, at the end of the second - 7 m/s, at the end of the third 9 m/s, etc. Obviously, . But how did we determine? We are looking at the speed difference over one second. In the first second 5-3=2, in the second second 7-5=2, in the third 9-7=2. But what if the speeds are not given for every second? Such a problem: the initial speed of the ship is 3 m/s, at the end of the second second - 7 m/s, at the end of the fourth 11 m/s. In this case, you need 11-7 = 4, then 4/2 = 2. We divide the speed difference by the time interval.

This formula is most often used in a modified form when solving problems:

The formula is not written in vector form, so we write the “+” sign when the body is accelerating, the “-” sign when it is slowing down.

Acceleration vector direction

The direction of the acceleration vector is shown in the figures

In this figure, the car moves in a positive direction along the Ox axis, the velocity vector always coincides with the direction of movement (directed to the right). When the acceleration vector coincides with the direction of the speed, this means that the car is accelerating. Acceleration is positive.

During acceleration, the direction of acceleration coincides with the direction of speed. Acceleration is positive.

In this picture, the car is moving in the positive direction along the Ox axis, the velocity vector coincides with the direction of movement (directed to the right), the acceleration does NOT coincide with the direction of the speed, this means that the car is braking. Acceleration is negative.

When braking, the direction of acceleration is opposite to the direction of speed. Acceleration is negative.

Let's figure out why the acceleration is negative when braking. For example, in the first second the motor ship dropped its speed from 9m/s to 7m/s, in the second second to 5m/s, in the third to 3m/s. The speed changes to "-2m/s". 3-5=-2; 5-7=-2; 7-9=-2m/s. This is where the negative acceleration value comes from.

When solving problems, if the body slows down, acceleration is substituted into the formulas with a minus sign!!!

Moving during uniformly accelerated motion

An additional formula called timeless

Formula in coordinates

Medium speed communication

With uniformly accelerated motion, the average speed can be calculated as the arithmetic mean of the initial and final speeds

From this rule follows a formula that is very convenient to use when solving many problems

Path ratio

If a body moves uniformly accelerated, the initial speed is zero, then the paths traversed in successive equal intervals of time are related as a successive series of odd numbers.

The main thing to remember

1) What is uniformly accelerated motion;
2) What characterizes acceleration;
3) Acceleration is a vector. If a body accelerates, the acceleration is positive, if it slows down, the acceleration is negative;
3) Direction of the acceleration vector;
4) Formulas, units of measurement in SI

Exercises

Two trains are moving towards each other: one is heading north at an accelerated rate, the other is moving slowly to the south. How are train accelerations directed?

Equally to the north. Because the first train's acceleration coincides in direction with the movement, and the second train's acceleration is opposite to the movement (it slows down).

Acceleration of a point during linear motion

Mechanical movement. Basic concepts of mechanics.

Mechanical movement– change in the position of bodies (or their parts) in space over time relative to other bodies.

From this definition it follows that mechanical movement is movement relative.

The body in relation to which this mechanical movement is considered is called body of reference.

Frame of reference- this is a set of reference body, coordinate system and time reference system associated with this body, in relation to which the movement (or equilibrium) of any other material points or bodies is studied(Fig. 1).

Inertial reference frame(i.s.o.)– a reference system in which the law of inertia is valid: a material point, when no forces act on it (or mutually balanced forces act on it), is in a state of rest or uniform linear motion.

Any frame of reference moving with respect to And. With. O. progressively, evenly and rectilinearly, there is also And. With. O. Therefore, theoretically there can be any number of equal rights And. With. O., which have the important property that in all such systems the laws of physics are the same (the so-called principle of relativity).

If the reference system moves relative to the i.s.o. unevenly and rectilinearly, then it is non-inertial and the law of inertia is not fulfilled in it. This is explained by the fact that in relation to a non-inertial reference system, a material point will have acceleration even in the absence of acting forces, due to the accelerated translational or rotational motion of the reference system itself.

The concept of i. With. O. is a scientific abstraction. The real reference system is always associated with some specific body (the Earth, the hull of a ship or aircraft, etc.), in relation to which the movement of certain objects is studied. Since there are no motionless bodies in nature (a body motionless relative to the Earth will move with it accelerated relative to the Sun and stars, etc.), then any real reference system is non-inertial and can be considered as And. With. O. only to varying degrees of approximation.

With a very high degree of accuracy And. With. O. can be considered a so-called heliocentric (stellar) system with the beginning at the center of the Sun (more precisely, at the center of mass of the Solar system) and with axes directed towards three stars. To solve most technical problems And. With. O. In practice, a system rigidly connected to the Earth can serve, and in cases requiring greater accuracy (for example, in gyroscopy), with a beginning in the center of the Earth and axes directed to the stars.

When moving from one And. With. O. to the other, in classical Newtonian mechanics, the Galilean transformations are valid for spatial coordinates and time, and in relativistic mechanics (i.e., at speeds of movement close to the speed of light) the Lorentz transformations are valid.

Material point– a body whose dimensions, shape and internal structure can be neglected in the conditions of this problem.

A material point is an abstract object.

Absolutely solid body(ATT) – a body, the distance between any two points of which remains unchanged (the deformation of the body can be neglected).

ATT is an abstract object.

Finite movement – ​​movement in a limited area of ​​space, infinite movement – ​​movement unlimited in space.

Point position A in space the radius is specified - by a vector or its three projections on the coordinate axes (Fig. 2).

Fig.2.

Kinematics

Kinematics– a branch of mechanics devoted to the study of the laws of motion of bodies without taking into account their masses and acting forces.

Basic concepts of kinematics

Path – scalar physical quantity equal to the length of the trajectory section, traveled by a material point during the period of time under consideration; in SI: = m(meter).

In classical physics it was implicitly assumed that the linear dimensions of a body are absolute, i.e. are the same in all inertial reference systems. However, in the special theory of relativity it proves length relativity(reduction of the linear dimensions of the body in the direction of its movement).

The linear dimensions of a body are greatest in the frame of reference relative to which the body is at rest:Δ l =Δ i.e. > , where is the body’s own length, i.e. body length measured in ISO, relative to which the body is at rest, where .

Moving –vector,connecting the position of a moving point at the beginning and end of a certain period of time(Fig. 6); in SI: .

Fig.6.

– movement, ABCD- path. Fig.7.

Example. The motion of a point is given by the equations:

Write an equation for the trajectory of the point and determine its coordinates after the start of the movement.

Fig.8.

The initial position of the point (at ) is determined by the coordinates, cm. In 1 sec. the point will be in position with coordinates:

Time(t) – one of the categories(along with space), denoting the form of existence of matter; form of physical and mental processes; expresses the order of change of phenomena; condition for the possibility of change, as well as one of the space coordinates–time along which the world lines of physical bodies are stretched; in SI: – second.

In classical physics it was implicitly assumed that time is an absolute value, i.e. the same in all inertial reference systems. However, in the special theory of relativity, the dependence of time on the choice of inertial reference system was proven: , where is the time measured by the watch of an observer moving with the reference system. This led to the conclusion that relativity of simultaneity, namely: in contrast to classical physics, where it was assumed that simultaneous events in one inertial frame of reference are simultaneous in another inertial frame of reference, in the relativistic case spatially separated events that are simultaneous in one inertial frame of reference may be non-simultaneous in another frame of reference.

H.2. Speed

Speed(often denoted , or from English. velocity or fr. vitesse)– vector physical quantity characterizing the speed of movement and direction of movement of a material point in space relative to the selected reference system.

Instantaneous speed – vector quantity equal to the first derivative of the radius of the vector moving point in time(the speed of a body at a given time or at a given point on the trajectory):

The instantaneous velocity vector is directed tangentially to the trajectory in the direction of the point’s movement (Fig. 9).

Rice. 9.

In the same time , That's why

Thus, the coordinates of the velocity vector are the rates of change of the corresponding coordinate of the material point:

or in notation:

Then the velocity module can be represented: In general, the path is different from the displacement module. However, if we consider the path traversed by a point in a short period of time , That . Therefore, the magnitude of the velocity vector is equal to the first derivative of the path length with respect to time: .

If the modulus of the point velocity does not change over time , then the movement is called uniform.

For uniform motion the following relation is valid: .

If the velocity modulus changes with time, then the motion is called uneven.

Uneven motion is characterized by average speed and acceleration.

The average ground speed of the uneven movement of a point in a given section of its trajectory is called a scalar quantity , equal to the ratio of the length of this section, trajectory to the duration of time passing it as a point(Fig. 10): , where is the path traveled by the point in time .

Rice. 10. Vectors of instantaneous and average speed.

Rice. eleven.

In classical mechanics, speed is a relative quantity, i.e. is transformed upon transition from one inertial reference system to another according to Galilean transformations.

When considering complex motion (that is, when a point or body moves in one reference system, and the reference system itself moves relative to another), the question arises about the connection of velocities in 2 reference systems, which establishes the classical law of addition of velocities:

the speed of a body relative to a stationary frame of reference is equal to the vector sum of the speed of the body relative to a moving frame and the speed of the moving system itself relative to a stationary frame:

where is the speed of a point relative to a stationary reference system, is the speed of a moving reference system relative to a stationary system, is the speed of a point relative to a moving reference system.

Example:

1. The absolute speed of a fly crawling along the radius of a rotating gramophone record is equal to the sum of the speed of its movement relative to the record and the speed that the point of the record under the fly has relative to the ground (that is, with which the record carries it due to its rotation).

2. If a person walks along the corridor of a carriage at a speed of 5 kilometers per hour relative to the carriage, and the carriage is moving at a speed of 50 kilometers per hour relative to the Earth, then the person moves relative to the Earth at a speed of 50 + 5 = 55 kilometers per hour when walking in the direction of movement train, and at a speed of 50– 5 = 45 kilometers per hour when it goes in the opposite direction. If a person in a carriage corridor moves relative to the Earth at a speed of 55 kilometers per hour, and a train at a speed of 50 kilometers per hour, then the speed of the person relative to the train is 55–50 = 5 kilometers per hour.

3. If the waves move relative to the shore at a speed of 30 kilometers per hour, and the ship also at a speed of 30 kilometers per hour, then the waves move relative to the ship at a speed of 30 – 30 = 0 kilometers per hour, that is, relative to the ship they become motionless.

In the relativistic case, the relativistic law of velocity addition is applied: .

From the last formula it follows that the speed of light is the maximum speed of transmission of interactions in nature.

Acceleration

Acceleration is a quantity that characterizes the rate of change in speed.

Acceleration(usually denoted ) – derivative of speed with respect to time, a vector quantity showing how much the speed vector of a point (body) changes as it moves per unit time(i.e. acceleration takes into account not only the change in the magnitude of the speed, but also its direction).

For example, near the Earth, a body falling on the Earth, in the case where air resistance can be neglected, increases its speed by approximately 9.81 m/s every second, that is, its acceleration, called the acceleration of gravity .

Derivative of acceleration with respect to time, i.e. the quantity characterizing the rate of change of acceleration is called jerk.

The acceleration vector of a material point at any time is found by differentiating the velocity vector of the material point with respect to time:

.

The acceleration module is an algebraic quantity:

- movement accelerated(speed increases in magnitude);

- movement slow(speed decreases in magnitude);

– uniform movement.

If – movement equally variable(uniformly accelerated or equally decelerated).

Average acceleration

Average acceleration is the ratio of the change in speed to the period of time during which this change occurred:

Where - vector of average acceleration.

The direction of the acceleration vector coincides with the direction of the change in speed (here this is the initial speed, that is, the speed at which the body began to accelerate).

At the moment of time the body has speed. At the moment of time, the body has a speed (Fig. 12). According to the rule of subtraction of vectors, we find the vector of change in speed. Then you can determine the acceleration like this:

Rice. 12.

Instant acceleration.

Instantaneous acceleration of a body (material point) at a given moment in time is a physical quantity equal to the limit to which the average acceleration tends as the time interval tends to zero. In other words, this is the acceleration that the body develops in a very short period of time:

.

The direction of acceleration also coincides with the direction of the change in speed for very small values ​​of the time interval during which the change in speed occurs.

The acceleration vector can be specified by projections onto the corresponding coordinate axes in a given reference system:

those. the projection of the acceleration of a point onto the coordinate axes is equal to the first derivatives of the velocity projections or the second derivatives of the corresponding coordinates of the point with respect to time. The module and direction of acceleration can be found from the formulas:

,

where are the angles formed by the acceleration vector with the coordinate axes.

Acceleration of a point during linear motion

If the vector , i.e. does not change with time, the movement is called uniformly accelerated. For uniformly accelerated motion, the formulas are valid:

With accelerated linear motion, the speed of the body increases in absolute value, that is, and the direction of the acceleration vector coincides with the velocity vector , (i.e.).

Rice. 13.

Acceleration of a point during curvilinear motion

When moving along a curved path, not only the speed module changes, but also its direction. In this case, the acceleration vector is represented as two components.

Indeed, when a body moves along a curved path, its speed changes in magnitude and direction. The change in the velocity vector over a certain short period of time can be specified using a vector (Fig. 14).

The vector of changes in speed in a short time can be decomposed into two components: , directed along the vector (tangential component), and , directed perpendicular to the vector (normal component).

Then the instantaneous acceleration is: .

Tangential (tangential) acceleration – this is the component of the acceleration vector directed along the tangent to the trajectory at a given point of the motion trajectory. Tangential acceleration characterizes the change in speed modulo during curvilinear motion:

Normal(centripetal) acceleration

Normal acceleration is the component of the acceleration vector directed along the normal to the trajectory of motion at a given point on the trajectory of the body. That is, the normal acceleration vector is perpendicular to the linear speed of movement (Fig. 15). Normal acceleration characterizes the change in speed in direction and is indicated by the symbol. The normal acceleration vector is directed along the radius of curvature of the trajectory. From Fig. 15 it is clear that

Rice. 17. Movement along circular arcs.

Normal acceleration depends on the magnitude of the velocity and on the radius of the circle along the arc of which the body is moving at the moment.